The proof of Lemma REF  can be found in Xin [1]}.
However, the description of Lemma REF  is equivalent to but a
little different from Xin [1]}. Take the estimation
(REF ) as an example, which is described by
\(\Vert \partial _x w^r_\delta (\cdot ,t)\Vert _{L^p}\le \displaystyle C\min \lbrace (w_+-w_-)\delta ^{-1+1/p}, (w_+-w_-)^{1/p}t^{-1+1/p}\rbrace ,\qquad \mathrm {{(\ref {w1})´}}\) 
